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Feedback control of chaotic systems using Multiple Shooting Shadowing and
application to Kuramoto Sivashinsky equation
File(s)
main.pdf (1.13 MB)
Accepted version
Author(s)
Papadakis, Georgios
Shawki, Karim
Type
Journal Article
Abstract
We propose an iterative method to evaluate thefeedback control kernel of a chaotic system directlyfrom the system’s attractor. Such kernels are currentlycomputed using standard linear optimal controltheory, known as Linear Quadratic Regulator (LQR)theory. This is however applicable only to linearsystems, which are obtained by linearising thesystem governing equations around a target state.In the present paper, we employ the PreconditionedMultiple Shooting Shadowing (PMSS) algorithm tocompute the kernel directly from the non-linear dynamics, thereby bypassing the linear approximation.Using the adjoint version of the PMSS algorithm,we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto Sivashinsky equation. We compare the computed kernel with that produced by the standard LQR algorithm and note similarities and differences. Bothkernels are stabilising, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.
Date Issued
2020-08-26
Date Acceptance
2020-07-17
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020, 476 (2240), pp.1-20
URI
URL
DOI
ISSN
1364-5021
Publisher
The Royal Society
Start Page
1
End Page
20
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
476
Issue
2240
Copyright Statement
© 2020 The Author(s)

Published by the Royal Society. All rights reserved.
Identifier
https://royalsocietypublishing.org/doi/10.1098/rspa.2020.0322
Subjects
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published
Date Publish Online
2020-08-19